Sliding Mode Control For Systems With Slow and Fast Modes

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c 2010

THANG TIEN NGUYEN

ALL RIGHTS RESERVED


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ABSTRACT OF THE DISSERTATION

Sliding Mode Control for Systems with Slow and Fast Modes

by THANG TIEN NGUYEN

Dissertation Director:

Professor Zoran Gajic

This dissertation addresses the problems of sliding mode control for systems with slow

and fast dynamics. Sliding mode control is a type of variable structure control, where

sliding surfaces or manifolds are designed such that system trajectories exhibit de-

sirable properties when confined to these manifolds. A system using a sliding mode

control strategy can display a robust performance against parametric and exogenous

disturbances under the matching condition (Drazenovics condition). This property

is of extreme importance in practice where most systems are affected by parametric

uncertainties and external disturbances.

First, we investigate a high gain output feedback sliding mode control problem for

sampled-data systems with an unknown external disturbance. It is well-known that

under high gain output feedback, a regular system can be brought into a singularly

perturbed form with slow and fast dynamics. An output feedback based sliding surface

is designed using some standard techniques for continuous-time systems. Next, we con-

struct a discrete-time output feedback sliding mode control law for the sliding surface.

The main challenge in this work is the appearance of the external disturbance in the

control law. A remedy is to approximate the disturbance by system information of the

previous time sampling period. The synthesized control law is able to provide promising

results with high robustness against the external disturbance, which is demonstrated

by the bounds of the sliding mode and state variables. These characteristics are further

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improved by a method which takes into account system information of two previous

time instants in order to better approximate the disturbance. The stability and ro-

bustness of the closed-loop system under the proposed control laws are analyzed by

studying a transformed singularly perturbed discrete-time system.

The second topic of the thesis is to study sliding mode control for singularly per-

turbed systems which exhibit slow and fast dynamics. A state feedback control law

is designed for either slow or fast modes. Then, the system under that state feedback

control law is put into a triangular form. In the new coordinates, a sliding surface

is constructed for the remaining modes using Utkin and Youngs method. The sliding

mode control law is synthesized by a control method which is an improved version of the

unit control method by Utkin. Lastly, the proposed composite control law consisting

of the state feedback control law and sliding mode control is realized. It is shown that

stability and disturbance rejection are achieved. Our results show much improvement

when compared to the other works available in the literature on the same problem.

The problem of sliding mode control for singularly perturbed systems is also ad-

dressed by the Lyapunov approaches. First, a state feedback composite control is

designed to stabilize the system. Then, Lyapunov functions based on the state feed-

back control law and the system dynamics are employed in an effort to synthesize a

sliding surface. Two sliding surfaces and two sliding mode controllers are proposed in

this direction. Theoretical and simulation results show the effectiveness of the proposed

methods. Like composite approaches, the Lyapunov ones provide asymptotic stability

and disturbance rejection.

We also study singularly perturbed discrete-time systems with parametric uncer-

tainty. Proceeding along the same lines as in the continuous-time case, we propose two

approaches to construct a composite control law: a state feedback controller to stabi-

lize either slow or fast modes and a sliding mode controller designed for the remaining

modes. It is shown that the closed-loop system under the proposed control laws is

asymptotically stable provided the perturbation parameter is small enough.

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Acknowledgements

The writing of the dissertation could not have been complete without the enormous

support of numerous people. I would like to acknowledge all those people who have

made contribution to the completion of this dissertation.

First, my great thanks go to my advisor, Prof. Zoran Gajic who has spared no time

and energy to teach and advise me for years. His profound ideas and guidance have

helped me to learn how to do research and write scientific papers. I will cherish all of

experiences that I gain when working with him. In addition, I deeply acknowledge Prof.

Gajics wife, Dr. Verica Radisavljevic for numerous discussions and lectures related to

my study.

I also would like to extend my warm gratitude to Prof. Wu-Chung Su from National

Chung-Hsing University, Taiwan, my doctoral dissertation co-advisor. His continuous

guidance and encouragement lead me to learn sliding mode control and face with in-

teresting problems which play a pivotal role in my dissertation.

I am greatly grateful Prof. Dario Pompili, Prof. Predrag Spasojevic who serve as

committee members for their insightful comments and constructive criticism that help

finish the dissertation.

I am also indebted to many people on the faculty and staff of the Department of

Electrical and Computer Engineering. Particularly, I would like to extend my thanks

to Prof. Yicheng Lu and Ms. Lynn Ruggiero for their support and practical advice to

pursue my studies in the department. I am especially thankful to Prof. Narinda Puri,

Prof. Sophocles Osfanidis, Prof. Eduardo Sontag, Prof. Roy Yates for inspiring me to

learn and explore new areas.

Several friends have been with me through these challenging but rewarding years.

Their concern and support have helped me overcome obstacles and stay focused on my

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direction. I greatly appreciate their friendship and assistance. Bo-Hwan Jung, Gun-

Hyung Park, Prashanth Gopala, Maja Skataric, Meng-Bi Cheng, Phong Huynh, Tuan

Nguyen, and many others.

Most importantly, my dissertation would not have been possible without the en-

couragement and support from my family. I would like to dedicate my dissertation to

my parents and my siblings who are a source of care, love, support and strength during

my graduate study.

Finally, I deeply appreciate financial support from the Vietnam Education Founda-

tion and the Department of Electrical Engineering that have funded my research and

study at Rutgers University.

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Table of Contents

Abstract . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ii

Acknowledgements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . iv

List of Figures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ix

1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1

1.1. Sliding Mode Control . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1

1.2. Singularly Perturbed Systems . . . . . . . . . . . . . . . . . . . . . . . . 5

1.3. Literature on Relevant Works . . . . . . . . . . . . . . . . . . . . . . . . 9

1.4. Contributions of the Dissertation . . . . . . . . . . . . . . . . . . . . . . 13

1.5. Organization of the Dissertation . . . . . . . . . . . . . . . . . . . . . . 14

2. Output Feedback Sliding Mode Control for Sampled-Data Systems 16

2.1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16

2.2. Problem Formulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18

2.3. Discrete-time Regular Form . . . . . . . . . . . . . . . . . . . . . . . . . 19

2.4. One-Step Delayed Disturbance Approximation Approach . . . . . . . . . 24

2.4.1. Output Feedback Control Design . . . . . . . . . . . . . . . . . . 24

2.4.2. Stability Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . 26

2.4.3. Accuracy Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . 31

2.5. Two-Step Delayed Disturbance Approximation Approach . . . . . . . . 34

2.5.1. Output Feedback Control Design . . . . . . . . . . . . . . . . . . 34

2.5.2. Stability Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . 34

2.5.3. Accuracy Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . 39

2.6. Numerical Example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42

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2.7. Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43

3. Sliding Mode Control for Singularly Perturbed Linear Continuous-

Time Systems: Composite Control Approaches . . . . . . . . . . . . . . . 47

3.1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47


3.3. Main Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57

3.3.1. Dominating Slow Dynamics Approach . . . . . . . . . . . . . . . 57

3.3.2. Dominating Fast Dynamics Approach . . . . . . . . . . . . . . . 62

3.4. Numerical Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66

3.4.1. Example 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66

3.4.2. Example 2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73

3.5. Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 79

4. Sliding Mode Control for Singularly Perturbed Linear Continuous-

Time Systems: Lyapunov Approaches . . . . . . . . . . . . . . . . . . . . . 80

4.1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 80


4.3. Main Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81



4.4. Numerical Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 88

4.4.1. Example 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 88

4.4.2. Example 2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 89

4.5. Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 97

5. Sliding Mode Control for Singularly Perturbed Linear Discrete-Time

Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 100

5.1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 100


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5.3. Main Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102



5.4. Numerical Example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1095.5. Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 117

6. Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 118

6.1. Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 118

6.2. Future Work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 119

Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 121

Vita . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 127

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List of Figures

2.1. Evolution of the control law for the one-step delayed disturbance approx-

imation approach . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43

2.2. O() bounds of the state variables for the one-step delayed disturbance

approximation approach . . . . . . . . . . . . . . . . . . . . . . . . . . . 44

2.3. O(2) accuracy of the sliding motion for the one-step delayed disturbance


2.4. Evolution of the control law for the two-step delayed disturbance approx-

imation approach . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45

2.5. O(2) bounds of the state variables for the two-step delayed disturbance


2.6. O(3) accuracy of the sliding motion for the two-step delayed disturbance


3.1. Evolution of the slow state variables for the dominating slow dynamics

approach . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68

3.2. Evolution of the fast state variables for the dominating slow dynamics

approach . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68

3.3. Sliding function evolution for the dominating slow dynamics approach . 69

3.4. Evolution of the composite control law for the dominating slow dynamics

approach . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70

3.5. Evolution of the slow state variables for the dominating fast dynamics

approach . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71

3.6. Evolution of the fast state variables for the dominating fast dynamics

approach . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71

3.7. Evolution of the sliding function for the dominating fast dynamics approach 72

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3.8. Evolution of the composite control law for the dominating fast dynamics

approach . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72


approach . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74


approach . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75


3.12. Evolution of the composite control law for the dominating slow dynamics

approach . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 76


approach . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 773.14. Evolution of the fast state variables for the dominating fast dynamics

approach . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 78

3.15. Evolution of the sliding function for the dominating fast dynamics approach 78


approach . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 79


approach . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 90


approach . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 90


4.4. Evolution of the sliding mode control law for the dominating slow dy-

namics approach . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91


approach . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92

4.6. Evolution of the fast state variables for the dominating fast dynamics

approach . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92

4.7. Sliding function evolution for the dominating fast dynamics approach . 93

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4.8. Evolution of the sliding mode control law for the dominating fast dy-

namics approach . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93


approach . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 95


approach . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 95


4.12. The evolution of the sliding mode control law for the dominating slow

dynamics approach . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 96


approach . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 974.14. Evolution of the fast state variables for the dominating fast dynamics

approach . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 98

4.15. Sliding function evolution for the dominating fast dynamics approach . 98

4.16. Evolution of the sliding mode control law for the dominating fast dy-

namics approach . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 99


approach. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 111


approach. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 112

5.3. Sliding function evolution for the dominating slow dynamics approach. . 113

5.4. Evolution of the composite control law for the dominating slow dominat-

ing approach. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 113


approach. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 115

5.6. Evolution of the fast state variable for the dominating fast dynamics

approach. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 115

5.7. Sliding function evolution for the dominating fast dynamics approach. . 116

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approach. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 116

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1

Chapter 1

Introduction

1.1 Sliding Mode Control

Variable structure systems (VSS) are special structure systems that have been exten-

sively studied for decades. The basic philosophy of the variable structure approach is

that the structure of the system varies under certain conditions from one to anothermember of a set of admissible continuous time functions (Utkin, 1977). A VSS can

inherit combined useful properties from the structures. In addition, it can be endowed

with special properties which are not present in any of the structures (Utkin, 1977).

Sliding mode control is a type of variable structure control, where sliding surfaces

or manifolds are designed such that system trajectories exhibit desirable properties as

confined to these manifolds. A system using a sliding mode control strategy can display

a robust performance against parametric uncertainties and exogenous disturbances.This property is of extreme importance in practice where most of plants are heavily

affected by parametric and external disturbances.

Consider a general VSS described by

x(t) = f(x(t), t , u(t)) (1.1)

where x(t) Rn, and u(t) Rm. Each component of control is assumed to act indiscontinuous fashions based on some appropriate conditions,

ui(t) =

u+i (x, t) if si(x) > 0

ui (x, t) if si(x) < 0

, i = 1,...,m (1.2)

where si(x) plays the role of a sliding surface.

Since differential equations (1.1), (1.2) have discontinuous right hand sides, they do

not meet the classical requirements on the existence and uniqueness of solutions. A


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formal technique called the equivalent control method was introduced by Utkin (1977)

to analyze the equivalent dynamics of the closed-loop system. In this approach, a

control called the equivalent control is obtained by solving s(t) = 0, namely

ds

dt =s

x .dx

dt =s

x f(x, ueq, t) = 0. (1.3)

The system dynamics on the sliding surface is studied by substituting the equivalent

control ueq in equation (1.1). Note that the equivalent control ueq is not physically real-

izable due to the unknown disturbances. Furthermore, the equivalent system dynamics

is not exactly but very close to the sliding dynamics (Utkin, 1977).

The existence of the sliding mode is described by the following conditions (Utkin,

1978)

Sliding condition (sufficient, local)

lims0+

s < 0, lims0

s > 0 (1.4)

Reaching condition (sufficient, global)

s < sgn(s), (1.5)

where > 0 is a parameter to be designed.

Control magnitude constraint (necessary)

umin ueq umax (1.6)

In sliding mode control, a sliding surface is first constructed to meet existence con-

ditions of the sliding mode. Then, a discontinuous control law is sought to drive the

system state to the sliding surface in a finite time and stay thereafter on that surface.

We now present some fundamental designs of sliding mode control for regular linear

systems. Consider a linear system

x(t) = Ax(t) + Bu(t) + Df(t) (1.7)

where x(t) Rn is the state, u(t) Rm is the control, f(t) Rr is the unknown butbounded exogenous disturbance f M, with m p < n. It is assumed that B is


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of full rank, (A, B) is controllable and the following matching condition (Drazenovic,

1969) is satisfied

rank[B|D] = rankD (1.8)

In other words, D can be written as D = BG.

There are several systematic methods for constructing sliding surfaces, such that

Utkin and Youngs method (Utkin and Young, 1979), Lyapunov method (Gutman and

Leitmann, 1976; Gutman, 1979; Su et al., 1996). Utkin and Youngs method and

Lyapunov method to be presented below will be employed in solving our problem in

the following chapters.

Since B has full rank, there exists a nonsingular transformation T such that

T B =

0(nm)m

B0

,

x1(t)

x2(t)

= T x(t),

which bring (1.7) into the normal form

x1(t)

x2(t)

=

A11 A12

A21 A22

x1(t)

x2(t)

+

0

B0

u(t) +

0

B0G

f(t) (1.9)

where x1(t) Rnm, x2(t) Rm. Note that B0 is an mm matrix and it is nonsingular

because B is of full rank.

Regard x2(t) as a control input to the first subsystem of (1.9)

x1(t) = A11x1(t) + A12x2(t) (1.10)

and construct a state feedback gain for (1.10) as

x2(t) = Kx1(t) (1.11)

Hence, the sliding surface in the (x1, x2) coordinate can be chosen as

[K Imm]

x1

x2

= 0 (1.12)

or

s(t) = Cx(t) = [K Imm]T x(t) = 0 (1.13)


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in the original coordinates. Once the sliding surface coefficient matrix C is designed,

one can proceed to construct the sliding mode control law. Taking the derivative of

(1.13) with respect to t, we have

s(t) = Cx(t) = CAx(t) + CBu(t) + CDf(t) (1.14)

It is important to emphasize that the matrix CB in (1.14) is an m m nonsingularmatrix equal to B0 since

CB = [K I]T B = [K I]

0

B0

= B0. (1.15)

A sliding mode control law can be designed by using the unit control method or the

signum method (Utkin, 1984). From (1.15), a unit control law can be chosen as (Utkin,1984)

u(t) = (CB )1CAx (CB )1(+ ) ss (1.16)

where

= CDM (1.17)

is a value that helps to tackle disturbances. It can be shown that (1.16) satisfies the

vector form of the reaching condition, that is

sTs = s s + sTCDd(t) < s (1.18)

where is chosen as in (1.17) and is a design parameter for adjusting the reaching

time. One can find the finite reaching time by considering the Lyapunov function

V = sTs (1.19)

Taking its derivative, we have

V < 2s = 2V (1.20)

This yields

dVV

< 2dt. (1.21)

Hence, V(t)

V(0) < t. (1.22)


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Let t = Tr be the time to reach the sliding mode (V(Tr) = 0). Thus, the reaching time

is bounded as

Tr 0. (1.24)

A sliding surface is chosen as

s(t) = HB

T

P x(t) = 0. (1.25)

where H is an m m nonsingular matrix. It was proven that the system (1.7) withthe sliding mode on the sliding surface (1.25) is asymptotically stable (Su et al., 1996).

The idea of employing Lyapunovs second method to construct a sliding surface can be

extended to nonlinear systems (Su et al., 1996).

1.2 Singularly Perturbed Systems

Singularly perturbed systems are systems that possess small time constant, or similar

parasitic parameters which usually are neglected due to simplified modeling. When

taking into account those small quantities, the order of the model is increased and the

computation needed for control design can be expensive and even ill-conditioned. How-

ever, if one uses a simplified model to design a control strategy, the desired performance

may not be achieved or the system can be unstable. As a result, singular perturbation

methods have been developed for years to address the stability and robustness of those

systems. For an extensive study, we refer to (Kokotovic et al., 1986; Gajic and Lim,

2001).

Consider a linear singularly perturbed system without control

x(t) = A11x(t) + A12z(t), x(t0) = x0

z(t) = A21x(t) + A22x(t), z(t0) = z0, (1.26)


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where x(t) and z(t) are respectively slow and fast state variables and is a small positive

parameter.

To analyze the system (1.26), one common way is to use the Chang transformation

to transform (1.26) into a block-diagonal system where the slow and fast dynamics are

completely decoupled (Chang, 1972). The Chang transformation is represented by x(t)

z(t)

=

I1 H

L I2 LH

(t)

(t)

= T1

(t)

(t)

(1.27)

and its inverse transformation is given by (t)

(t)

=

I1 HL H

L I2

x(t)

z(t)

= T

x(t)

z(t)

. (1.28)

where L and H matrices satisfy algebraic equations

A21 A22L + LA11 LA21L = 0 (1.29)

and

(A11 A12L)H H(A22 + LA12) + A12 = 0. (1.30)

Matrices L and H can be found using several methods. For example, the Newton

method is presented in (Grodt and Gajic, 1988). The resulting decoupled form is

(t)(t)

= A11 A12L 00 A22 + LA12

(t)(t)

. (1.31)

Now we present a short summary on the design of state feedback control for deter-

ministic linear continuous time singularly perturbed systems. Consider the following

controlled system

x(t) = A11x(t) + A12z(t) + B1u(t), x(t0) = x0

z(t) = A21x(t) + A22z(t) + B2u(t), z(t0) = z0. (1.32)

where x(t) Rn1, z(t) Rn2, and u(t) Rm. The system (1.33) is approximatelydecomposed into an n1 dimensional slow subsystem and an n2 fast subsystem by setting

= 0 in (1.32). The slow subsystem is

xs(t) = Asxs(t) + Bsus(t), xs(t0) = x0

zs(t) = A122 (A21xs(t) + B2us(t), (1.33)


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where

As = A11 A12A122 A21, Bs = B1 A12A122 B2 (1.34)

and the vectors xs(t), zs(t), and us(t) denote the slow parts of x(t), z(t) and u(t). The

fast subsystem is

zf(t) = A22zf(t) + B2uf(t), zf(t0) = z0 zs(t0), (1.35)

where zf(t) = z(t) zs(t), and uf(t) = u(t) us(t) describe the fast parts of thecorresponding variables z(t) and u(t). A composite control law consists of slow and fast

parts as

u(t) = us(t) + uf(t) (1.36)

where us(t) = G0xs(t), and uf(t) = G2zf(t) are independently constructed for the

slow and fast subsystems (1.33) and (1.35). G0 and G2 can be designed by using

classic control theory with an assumption that (As, Bs) and (A22, B2) is controllable.

Nonetheless, a realizable control law must be presented in terms of the actual system

states x(t) and z(t). Replacing xs(t) by x(t) and zf(t) by z(t)zf(t) bring the compositecontrol (1.34) into the realizable feedback form as follows.

u(t) = G0x(t) + G2[z(t) + A122 (A21x(t) + B2G0x(t))] = G1x(t) + G2z(t) (1.37)

where

G1 = (I1 + G2A122 B2)G0 + G2A

122 A21. (1.38)

The discrete-time version of singularly perturbed systems is described in (Litkouhi

and Khalil, 1985). Consider the difference equation

x1[k + 1]

x2[k + 1]

=

(I1 + A11) A12

A21 A22

x1[k]

x2[k]

+

B1

B2

u[k] (1.39)

where x1 Rn1 , x2 Rn2, u Rm, > 0 is a small parameter and det[I2 A22 ] = 0.As in the continuous version, the slow and fast parts of equation (1.39) without control

can be separated by a decoupling transformation (Litkouhi and Khalil, 1985). A control

law which consists of slow and fast components is in the form

u[k] = us[k] + uf[k]


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where uf[k] decays exponentially. To investigate the slow subsystem, we neglect uf[k].

The resulting equations are given by

x1[k + 1] = (I1 + A11)x1[k] + A12x2[k] + B1us[k] (1.40)

x2[k] = A21x1[k] + A22x2[k] + B2us[k]. (1.41)

From (1.40) and (1.41), the slow subsystem is defined by

xs[k + 1] = (I1 + As)xs[k] + Bsus[k] (1.42)

where

xs = x1, (1.43)

As = A11 + A12(I2 A22)1A21, (1.44)

Bs = B1 + A12(I2 A22)1B2. (1.45)

If the pair (As, Bs) is stabilizable in the continuous sense, i.e., every eigenvalue of As

which lies in the closed right-half complex plane is controllable (Litkouhi and Khalil,

1985), then a state feedback control law for us[k] is designed as us[k] = Fsxs[k] where

Fs is chosen such that

Re{(As + BsFs)} < 0. (1.46)

With this choice, the closed-loop slow subsystem system is asymptotically stable. This

shows that the actual design problem for the discrete-time slow subsystem is a contin-

uous one (Litkouhi and Khalil, 1985).

The fast subsystem is defined by assuming that the slow variables are constant

during the fast transient, i.e., x[k + 1] = x[k], and us[k + 1] = us[k]. From (1.41) and

(1.39), the fast subsystem is given by

xf[k + 1] = Afxf[k] + Bfuf[k] (1.47)

where xf = x2 x2, Af = A22, and Bf = B2. If the pair (Af, Bf) is stabilizable in thediscrete-time sense, i.e., every eigenvalue of Af which lies outside or on the unit circle

is controllable (Litkouhi and Khalil, 1985), then a state feedback control law for uf[k]

is given by uf[k] = Ffxf[k] where Ff is chosen such that

|(Af + BfFf)| < 1. (1.48)


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As a result, the closed-loop fast subsystem is asymptotically stable. A composite control

law is taken as the sum of the slow and fast control components

u[k] = Fsxs[k] + Ffxf[k] = Fsxs[k] + Ff(x2[k] x2[k])

= [Fs Ff(I2 Af)1(A21 + BfFs)]x1[k] + Ffx2[k]. (1.49)

It was shown that under the composite control law (1.49), the closed-loop full-order

system is asymptotically stable for sufficiently small . Like in the continuous case, a

stabilizing state feedback control law can be synthesized from slow and fast controllers

that are designed independently.

1.3 Literature on Relevant Works

It is pointed out that for a regular system, a sliding mode control strategy can reject

disturbances and produce a robust performance. In systems with slow and fast modes,

little work has been devoted to the study of sliding mode control (Yue and Xu, 1996;

Su, 1999).

Yue and Xu (1996) studied a singularly perturbed system as follows

x(t) = A11x(t) + A12z(t) + B1u(t) + B1f(t,x,z),

z(t) = A21x(t) + A22z(t) + B2u(t) + B2g(t,x,z), (1.50)

where x(t) Rn1, z(t) Rn2, and u(t) Rm. f(t,x,z), g(t,x,z) : R+Rn1 Rn2 R denote the parameter uncertainties and external disturbances. 0 < < 1 represents

the singular perturbation parameter. Furthermore, the disturbances f(t,x,z), g(t,x,z)

are assumed to satisfy the following inequalities:

|f(t,x,z) 1(x, z) = a0 + a1x + a2z

g(t,x,z 2(x, z) = b0 + b1x + b2z.In addition, they satisfy

f(t,x,z) g(t,x,z) x + z.

In their approach, a designed control law includes two continuous time state feedback

terms and a switching term. The objective of the two continuous-time terms is to


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stabilize the system as no disturbances are taken into account. Specifically, the control

law is in the form

u = Kx + K0 + w (1.51)

where K and K0 are designed such that As + BsK and A22 + B2K0 are stable, and w

is a switching term to be defined. Here, is a new state variable given by

= z + A122 (A21 + B2K)x. (1.52)

To choose w, Yue and Xu (1996) considered a Lyapunov function candidate as follows

V = xTP1x + TP2

where P1, P2 are positive definite solutions to the following Lyapunov equations

(As + BsK)TP1 + P1(As + BsK) = Q1

(A22 + B2K0)TP2 + P2(A22 + B2K0) = Q2.

A control law is chosen as

u = Kx K0 (b0 + b01x + b2)sgn(BT1 P1x + BT2 P2) (1.53)

where b0, b01, and b2 are acquired from the definition of disturbances f(t,x,z) and

g(t,x,z) and matrices A21, A22, K. With this control law, the trajectories x and

ultimately satisfy (Yue and Xu, 1996)

x O(), O().

Yue and Xu (1996) also proved the existence of the sliding motion for the sliding

surface s = BT1 P1x + BT2 P2 provided some condition are satisfied. They employed

Lyapunov functions to construct the sliding function and the sliding mode control law.

Although, their approach deals with disturbances and provides some certain robustcharacteristics, they only guarantee that the trajectories of the system stay in an O()

boundary of the origin. Furthermore, it is somewhat complicated to compute some

parameters for their control law.

Heck (1991) studied a singularly perturbed system in the form of (1.50) without any

disturbances. The full-order system is separated into slow and fast subsystems (1.33),


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(1.35). A sliding-mode controller is constructed for each subsystem. Specifically, a slow

sliding surface can be chosen as

ss(xs) = Ssxs = 0

and a fast sliding surface is taken as

sf(zf) = Sfzf = 0.

Corresponding slow and fast control laws are designed for these sliding surfaces. A

composition of these control law is then implemented for the full-order system. Stability

analysis was carried out by using the equivalent method of Utkin (1977). It was shown

that the reduced-order subsystems approximate the full-order system with accuracy

O(). If reaching conditions are satisfied for the reduced-order models and an additional

condition is met, the reaching conditions are satisfied for the full-order system (Heck,

1991). One draw back of Hecks approach is that the boundedness of the derivative of

the slow sliding mode control must be supposed. Furthermore, parameter uncertainties

and external disturbances were not taken into consideration. As a result, Hecks scheme

is limited in real applications.

Li et al. (1995a) also considered a singularly perturbed system in the form of (1.32).

Similarly to Hecks approach, the full-order system is first decomposed into slow and

fast subsystems. Then, slow and fast sliding mode controllers are designed for the

subsystems individually. The composite control consists of two terms

u = ueq + u

where ueq is the equivalent control for the full-order system and u is the switching

term which is the regulating control moment for the full-order system. A sigmoid

function is exploited to eliminate the chattering phenomenon. Although the switching

surface of the full-order system is decided by the fast switching surface of the reduced-

order system, when reverting back to the original coordinates, the slow sliding control

still exists in the composite one. Like Hecks method, their approach does not address

uncertainties and external disturbances.


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Su (1999) studied the problem of sliding surface design for the system (1.32). Like

(Heck, 1991; Li et al., 1995a), the full-order system is separated into slow and fast

subsystems. Then, stabilizing state feedback controllers are constructed individually

for each subsystem, leading to a composite control law (1.36). The closed-loop system

is transformed into an exact slow and an exact fast subsystems by using the Chang

transformation (Chang, 1972; Kokotovic et al., 1986). The exact subsystems in the

new coordinates are

= Ts

= Tf.

There exist positive definite matrices Ps and Pf such that

PsTs + TTs Ps = Qs, Qs > 0

PfTf + TTf Pf = Qf, Qf > 0.

Then, the sliding surface for the singularly perturbed system can be chosen as

s(x, z) =

B1

B2/

T

JT

Ps 0

0 Pf

J

x

z

= 0.

It was shown that (Su, 1999) if the sliding motion is achieved, the system is asymptot-

ically stable. However, a control strategy has not been provided to realize the sliding

motion.

There have been several approaches to deal with the problem of sliding mode control

for singularly perturbed systems. Many of them (Heck, 1991; Li et al., 1995a; Su, 1999)

do not address parameter uncertainties and external disturbances which are inherent

in many practical plants. Heck (1991); Li et al. (1995a) exploited the decomposition

of the full-order system into slow and fast subsystems in an effort to construct sliding

surfaces and sliding mode controls for each subsystems. Meanwhile, Su (1999) only

took advantage of the structure of the closed-loop system under the composite control

law, by which a sliding surface is designed. Although Yue and Xu (1996) dealt with

parameter uncertainties and external disturbances, their method is not able to reject

disturbances completely. Instead, the trajectories of the system are ensured in an O()


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bound of the origin. One common feature of the above approaches is that Lyapunov

functions are employed to construct sliding surfaces and sliding mode controls.

1.4 Contributions of the Dissertation

The contributions of the dissertation are summarized in the following:

Two methods for design of an output feedback sliding mode control law forsampled-data systems are proposed in Chapter 2. The first method employs the

one-step predictor ahead technique to approximate the external disturbance. The

second method offers better way to approximate the disturbance by using system

information from two previous time instants. The main feature of the proposed

methods is that the control law is high gain, which leads to singular perturbation

behavior of the closed-loop system. The characteristics of the closed-loop systems

are much better than the other works available in the control literature. In addi-

tion, they capture the same level as in the state feedback case. The results in this

topic have been presented at 2009 American Control Conference, 2010 American

Control Conference, and published in IEEE Transactions on Automatic Control.

The development of two composite design methods for a singularly perturbedsystem with external disturbances is used to design a sliding mode controller. A

state feedback control law for either slow or fast modes combines with a sliding

mode control law to constitute a composite control law. The closed-loop system

under the proposed methods is asymptotically stable and robust against exter-

nal disturbances in the sliding mode. This contribution has been accepted for

publication in Dynamics of Continuous, Discrete and Impulsive Systems journal.

Two approaches based on the Lyapunov equations are used to design a slidingmode controller for singularly perturbed systems with external disturbances. The

results obtained are comparable to those of the composite control approaches;

namely, disturbance rejection and asymptotic stability in the sliding mode are

attained. These features show the advantages of the proposed methods when


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compared to other works available in the literature. This contribution is submitted

for journal publication.

The formulation and analysis of a sliding mode control problem for a discrete-

time singularly perturbed systems is investigated. Two composite approaches are

proposed to deal with the stability and robustness of a system with parametric

uncertainties. A state feedback control law is constructed for either slow or fast

modes. The remaining modes are handled by a sliding mode control law. A

composite control law combining the two components provides the system with

asymptotic stability and robustness against modeling uncertainties.

1.5 Organization of the Dissertation

Throughout the dissertation, we deal with different issues of sliding mode control for

singularly perturbed systems: discrete- and continuous-time domains. Each chapter

presents unified techniques or tools to address specific problems.

In Chapter 2, we formulate and develop an output feedback control strategy for

sampled-data systems. Stability and robustness will be analyzed by using singular

perturbation techniques. Furthermore, it will be theoretically shown that the same

characteristics as in the state feedback case are maintained. At the end of the chapter,

the numerical simulation of an aircraft model illustrates the advantages of the proposed

method.

Chapter 3 presents two composite control approaches for the sliding mode control

for singularly perturbed systems with external disturbances. We show how to design a

sliding surface and a corresponding control law to effectively stabilize the system and

reject the external disturbances. Two numerical examples at the end of the chapter

demonstrate the effectiveness of the proposed methods.

The same problem is addressed in Chapter 4 by different approaches. We employ

Lyapunov functions and the Chang transformation to construct a sliding surface and a

sliding mode control law to stabilize the closed-loop system and reject external distur-

bances.


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The techniques of Chapter 3 are applied to singularly perturbed discrete-time sys-

tems with parametric uncertainties in Chapter 5. Two composite sliding mode control

strategies are presented to deal with the stability and robustness of the closed-loop

system. A numerical example of a discrete-time model of a steam power system is

provided to illustrate the efficacy of the methods.

Finally, Chapter 6 presents an overview of the results of the dissertation which is

followed by some future directions.


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Chapter 2

Output Feedback Sliding Mode Control for Sampled-Data

Systems

2.1 Introduction

The problem of output feedback sliding mode control with disturbances has been ex-

tensively studied for years (Zak and Hui, 1993; EL-Khazali and DeCarlo, 1995; Hecket al., 1995; Edwards and Spurgeon, 1995, 1998). It was pointed out that the problem

of choosing the desired poles of the sliding mode dynamics can be approached by using

the classical squared-down techniques (MacFarlane and Karcanias, 1976). In order

to attain a global attraction to the sliding surface, Heck et al. (1995) established nu-

merical methods that adjust the switching gain to compensate for the unknown state

and disturbance variables. Edwards and Spurgeon (1995) proposed a procedure to con-

struct a sliding surface based on the output information by taking advantage of thefact that the invariant zeros of a system appear in the dynamics of the sliding motion.

The remaining eigenvalues of the sliding mode dynamics can be chosen appropriately

in the framework of a static output feedback pole placement problem for a subsystem

(Zak and Hui, 1993; Edwards and Spurgeon, 1995).

In this chapter, we consider the output feedback sliding mode control for sampled-

data linear systems. It is well-known that the exact continual sliding motion cannot be

achieved in the discrete-time case due to the sample/hold effect (Milosavljevic, 1985).Specifically, the system trajectory only travels in a neighborhood of the sliding surface

forming a boundary layer (Su et al., 2000). Several approaches were proposed to address

the problem of discrete-time output feedback sliding mode control (Lai et al., 2007; Xu

and Abidi, 2008a,b). Some of them are devoted to sliding mode control of sampled-data

systems. Xu and Abidi (2008a) employed a disturbance observer and a state observer


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with an integral sliding surface to address the output tracking problem for sampled-

data systems. Under their proposed control approach, the stability of the closed-loop

system is guaranteed and the effect of external disturbances is reduced (Xu and Abidi,

2008b).

The deadbeat control strategy proposed in (Oloomi and Sawan, 1997) is able to

decouple external disturbances with an O() accuracy, where is the sampling pe-

riod. In (Su et al., 2000), an one-step delayed disturbance approximation approach

has been shown to be effective in dealing with disturbances that exhibit certain con-

tinuity characteristics. We shall exploit the continuity attribute of the state variables

and the disturbances by using similar ideas to deal with the similar estimation problem

encountered above.

We will present two dynamic output feedback discrete-time control strategies that

take into account the disturbance compensation as in (Su et al., 2000). In the first

approach, the estimation of the disturbance is based on its previous time instant value,

while the second approach employs two previous time instant values. Both methods

possess high gain output feedback control. It is pointed out that with high gain output

feedback control, the system exhibits the two-time scale behavior (Litkouhi and Khalil,

1985; Gajic and Lim, 2001; Oloomi and Sawan, 1997; Young et al., 1977). By using

singular perturbation analysis, we will study the stability of the closed-loop system and

the accuracy of the sliding mode. Specifically, we will show that the state trajectory will

be remained in an O(2) boundary layer of the sliding surface in the first approach and

an O(3) vicinity of the sliding surface in the second method. While the first approach

shares with (Xu and Abidi, 2008a) some characteristics such as the bound of sliding

motion and the ultimate bound of state variables, the second one presents stronger

results. In addition to a controller for implementation, the method in (Xu and Abidi,

2008a) is performed with two observers for state and disturbance estimation, while we

employ no observers. On the other hand, the method of Xu and Abidi (2008a) applies

to systems of relative degree higher than one, while our methods are limited to systems

with relative degree one.

Throughout the dissertation, {A} denotes the spectrum of matrix A, while Im


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stands for an identity matrix of order m. A vector function f(t, ) Rn is said to beO() over an interval [t1, t2] Kokotovic et al. (1986) if there exists positive constants k

and such that

f(t, )

k

[0, ],

t

[t1, t2]

where . is the Euclidean norm. Moreover, it is said to be O(1) over [t1, t2] if

f(t, ) k, t [t1, t2].

2.2 Problem Formulation

We consider a linear system described by

x0(t) = A0x0(t) + B0u(t) + D0f(t) (2.1)

y(t) = C0x0(t)

where x0(t) Rn is the system state, u(t) Rm is the control, y(t) Rp is thesystem output, f(t) Rr is the unknown but bounded exogenous disturbance, withm p < n. The system matrices A0, B0, C0, D0 are constant of appropriate dimensionswith magnitudes O(1). The following assumptions are made:

1. The system (2.1) has relative degree 1.

2. Matrices B0 and C0 have full rank.

3. (A0, B0, C0) is controllable and observable (EL-Khazali and DeCarlo, 1995).

4. The invariant zeros of system (2.1) are stable.

In addition, D0 satisfies the matching condition (Drazenovic, 1969)

rank([B0

|D0]) = rank(B0) (2.2)

In other words, there exists a matrix K such that

D0 = B0K. (2.3)

The sliding surface under consideration is

s(t) = Hy(t) = HC0x0(t) = 0, (2.4)


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where H is a full rank m p matrix, designed to render stable sliding dynamics. It isshown that the eigenvalues of the sliding mode dynamics include the invariant zeros of

the system (2.1) (Edwards and Spurgeon, 1998). One can place the remaining eigen-

values of the zero dynamics of the sliding surface (2.4) if the Davison-Kimura condition

(Davison and Wang, 1975) is satisfied (Edwards and Spurgeon, 1998). In the case the

Davison-Kimura condition is not satisfied, a dynamic compensator is constructed to

produce a tractable structure for the sliding surface design (Edwards and Spurgeon,

1998). Refer to (EL-Khazali and DeCarlo, 1995; Zak and Hui, 1993; Edwards and

Spurgeon, 1998) for design of H. Note that HC0B0 is nonsingular. Our objective is to

construct a discrete-time sliding mode controller given output sliding surface (2.4).

2.3 Discrete-time Regular Form

In this section, we will use several similarity transformations to facilitate system design

and analysis. Since rank(B0) = m, there exists a coordinate transformation T0 such

that

B = T0B0 =

0

B2

.

where B2 is a nonsingular square matrix of dimension m. The new variables are defined

as

x =

x1

x2

= T0x0.

The similarity transformation T0 brings the original system (2.1) into the regular

form (Utkin and Young, 1979)

x1(t) = A11x1(t) + A12x2(t)

x2(t) = A21x1(t) + A22x2(t) + B2u(t) + D2f(t), (2.5)

where D2 = B2K. The sliding surface (2.4) is now described as

s(t) = HC0x0(t) = HC0T10 x(t) = Cx(t) = C1x1(t) + C2x2(t) = 0, (2.6)

where

HC0T10 = C.


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The zero dynamics of the sliding mode is represented by the eigenvalues of matrix

Ac = A11A12C12 C1. Note that H has been chosen in (2.4) to render a sliding surfacecoefficient matric C such that C2 is invertible and Ac is stable (Utkin and Young, 1979).

Sampling the continuous-time system (2.5) with the sampling period results in the

following discrete-time model

x[k + 1] = x[k] + u[k] + d[k] (2.7)

where

= eA, =

0

eAdB,

d[k] =

0

eABK f((k + 1)

)d. (2.8)

The system matrices, and , of the sampled-data system (2.7) can be reformulated

by taking the Taylor series expansion as

= eA = I+ A +2

2!A2 + = I+ (A + A) = O(1) (2.9)

and

=

0

eAd B = (B + B) = O(), (2.10)

where

A =1

2!A2 +

3!A3 + = O(1) (2.11)

and

B =1

2!AB +

3!A2B + =

B1

B2

= O(1), (2.12)

where the dimensions of B1 and B2 are (n m) m and m m, respectively.

Furthermore, since B =

0

B2

, can be written as

=

2B1

B2

, (2.13)

where

B2 = B2 + B2. (2.14)


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Due to the sampling process, the disturbance in the sampled-data system (2.7) exhibits

unmatched components, as demonstrated by the following lemma, which is an enhanced

version of Abidi et al. (2007).

Lemma 2.1. If the first and second derivatives of the disturbance f(t) in (2.1) are bothwell defined and bounded, then

d[k] =

0

eABK f((k + 1) )d = Kf[k] + 2

Kv [k] + 3d[k],

d[k] d[k 1] = O(2),

d[k] 2d[k 1] + d[k 2] = O(3), (2.15)

where v(t) = df(t)/dt and d[k] is a bounded uncertainty.

Proof. Consider 0 < and express f((k + 1) ) as

f((k + 1) ) = f[k] +(k+1)k

v()d

= f[k] +

(k+1)k

(v[k] +

k

v()d)d

= f[k] + v[k]( ) +(k+1)k

k

v()dd (2.16)

with k < (k + 1) . Substituting (2.16) into the expression of d[k] yields

d[k] =0

eABK f((k + 1) )d=

0

eABK f[k]d +

0

eABK v[k]( )d

+

0

eABK

(k+1)k

k

v()ddd. (2.17)

Following the same steps as in Lemma 1 in Abidi et al. (2007), we obtain0

eABK f[k]d = Kf[k], (2.18)

0 e

A

BK v[k]( )d =

2 Kv [k] +

M v[k]

3

, (2.19)

where M is a constant matrix. Assume the second derivative off(t) is bounded by W,

namely v(t) W. Then, we have

(k+1)k

k

v()dd (k+1)k

k

Wdd

W( )2 < W 2. (2.20)


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Hence,

0

eABK

(k+1)k

k

v()ddd

0

usi(xs), if ssi(xs) 0(3.7)

where ssi = Csixs is the ith linear switching function, u+si and u

si are smooth functions

to be defined later. The equivalent control for the slow subsystem during sliding is

given by Utkin (1984)

ues = (CsB0)1CsA0xs. (3.8)

For the fast subsystem (3.5), a sliding surface is chosen as

sf = Cfzf = 0. (3.9)

The control is chosen in the form:

ufi(zf) =

u+fi(zf), if sfi(zf) > 0

ufi(zf), if sfi(zf) 0(3.10)

where sfi = Cfizf is the ith linear switching function, u+fi and u

fi are smooth functions

to be defined later. The equivalent control for the fast subsystem during sliding is given

by Utkin (1984)

uef = (CfB2)1CfA22zf. (3.11)

The control law for the full-order system is a composite of the slow and fast controls

as given by

u = us(xs) + uf(zf) (3.12)


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where us is defined in (3.7) and uf is defined in (3.10).

The structure of the slow control law (3.7) and the fast control one (3.10) can be

chosen by three methods (Heck, 1991). For a system

x = Ax + Bu (3.13)

the control law is given (Utkin, 1977)

ui = (mi=1

|xm| )sgn(si) (3.14)

where ui is the ith component of the control, si is the ith row vector of s(t). A control

law proposed by DeCarlo et al. (1988) can be used for the slow subsystem

u = (SB)1SAx (SB)1SGN(Sx) (3.15)

where SGN(y) is a vector-valued function with the ith component equal to sgn(yi). The

composite control law can be given by

u = SGN(Kxx + Kzz) (3.16)

where Kx, Kz are to be determined. The main contribution of Heck is that the problem

of sliding mode control for the full order system (3.1) is addressed by two reduced-order

problems for the slow and fast subsystems. The drawback of Hecks method is the

assumption ondzs)dx() to validate the fast model; hence, the generality of the method is

limited. Furthermore, external disturbances were not brought into the picture.

Li et al. (1995a) also considered a singularly perturbed system in the form of (3.1).

Similarly to Hecks approach, the full-order system is first decomposed into slow and

fast subsystems. Then, slow and fast sliding surfaces are designed for the subsystems

separately. The only difference is that Li et al. (1995a) proposed a non switching control

strategy. The control law for the slow subsystem is given by

us = ues + us (3.17)

where ues is defined in (3.8) and

us = (CsB0)1ssig(Csxs) sgn(Csxs) (3.18)


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where stands for the component-wise multiplication, s is a positive scalar, andsig(Csxs) is a vector function whose ith component is given by

sig(Csixs) =1 e|Csixs|1 + e|Csixs|

0. (3.19)

Similarly, the fast sliding mode control law is given by

uf = uef + uf (3.20)

where

uef = (CfB2)1CfA22zf

is equivalent control and

uf = (CfB2)1ssig(Cf) sgn(Cfzf)

The composite control is constructed from slow and fast control laws:

u = us(x) + uf(zf)

where us and uf are defined in (3.17) and (3.20). The control method proposed by Li

et al. (1995a) employs sigmoid functions to reduce the chattering phenomenon. Like

(Heck, 1991), they did not consider the problem of external disturbances beside the

closed-loop stability issue.

Following the same direction as in (Heck, 1991; Li et al., 1995a), Innocenti et al.

(2003) studied a class of singularly perturbed systems:

x(t) = A11x(t) + A12z(t), x(t0) = x0

z(t) = A21x(t) + A22z(t) + B2u(t), z(t0) = z0, (3.21)

where x(t) Rn1, z(t) Rn2, u(t) Rm, is a small positive parameter, A22 isnonsingular, B2 is of full rank, and (A22, B2) is controllable. It is seen that system

(3.21) is less general than system (3.1) when the control input only affects the system

through the dynamics ofz(t). Like in (Heck, 1991), they employed standard techniques

of singular perturbations to decompose system (3.21) into slow and fast subsystems.


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The slow model is given by

xs(t) = A0xs + B0us(xs)

zs = A122 (A21xs + B2us(xs)) (3.22)

where A0 = (A11 A12A122 A21) and B0 = A12A122 B2. The slow sliding surface ischosen as

ss = Csxs = 0.

The slow control law is given by

us(xs) = (CsB0)1(Ksss + Qss(ss) + CsA0xs) (3.23)

where Ks Rrr

, Qs Rrr

are positive definite diagonal matrices and s() : Rr

Rr is a vector function, whose ith component is given by

si(i) =i

|i| + si(3.24)

and si is a small positive quantity. The fast reduced model obtained is the same as in

(Heck, 1991; Li et al., 1995a):

dzfd

= A22zf + B2uf (3.25)

where zf = z zs is the fast component of z and uf = u us is the fast control. If thefast sliding surface is chosen as

sf = Cfzf = 0,

the fast sliding control law is given by

uf(zf) = (CfB2)1(Kfsf + Qff(sf) + CfA22zf). (3.26)

The composite control is synthesized from slow control (3.23) and fast control (3.26)

uc = us(x) + uf(zf).

To study the stability of dynamics of subsystems, Innocenti et al. (2003) employed a

state space decomposition to construct a quadratic Lyapunov function and a procedure

in (Kokotovic et al., 1986). Under their proposed scheme, the closed-loop system is


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proved to be globally stable with sufficiently small . Different from Heck (1991); Li

et al. (1995a), Innocenti et al. (2003) utilized a continuous-time control law instead

of discontinuous one. Hence, it helps avoid chattering phenomena. While external

disturbances are also not considered, the class of systems under consideration is less

general than in (Heck, 1991; Li et al., 1995a).

Unlike the above approaches, where no disturbances are taken into account, Yue

and Xu (1996) studied a singularly perturbed system of the form

x(t) = A11x(t) + A12z(t) + B1u(t) + B1f(t,x,z),

z(t) = A21x(t) + A22z(t) + B2u(t) + B2g(t,x,z), (3.27)

where x(t) Rn1

, z(t) Rn2

, and u(t) Rm

. 0 < 1 represents the singularperturbation parameter. f(x,z ,t), g(x,z ,t) : R+Rn1Rn2 R denote the param-eter uncertainties and external disturbances. Furthermore, the disturbances f(x,z ,t),

g(x,z ,t) are assumed to satisfy the following inequalities:

|f(x,z ,t) 1(x, z) = a0 + a1x + a2z

g(x,z ,t) 2(x, z) = b0 + b1x + b2z.

In addition, they satisfy

f(x,z ,t) g(x,z ,t) x + z.

In their approach, a designed control law includes two continuous time state feedback

terms and a switching term. The objective of the two continuous-time terms is to

stabilize the system as no disturbances are taken into account. Specifically, the control

law is in the form

u = Kx + K0 + w (3.28)

where K and K0 are designed such that As + BsK and A22 + B2K0 are stable, and w

is a switching term to be defined. Here, is a new state variable given by

= z + A122 (A21 + B2K)x. (3.29)


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The new system is given by

x = A011x + (A12 + B1K0) + B1w + B1f

= (A22 + B2K0) + B2w + B2g + O().

To choose w, Yue and Xu considered a Lyapunov function candidate as follows

V = xTP1x + TP2

where P1, P2 are positive definite solutions to the following Lyapunov equations

(As + BsK)TP1 + P1(As + BsK) = Q1

(A22 + B2K0)TP2 + P2(A22 + B2K0) =

Q2.

The control law is chosen as

u = Kx K0 (b0 + b01x + b2)sgn(BT1 P1x + BT2 P2) (3.30)

where b0, b01, and b2 are acquired from the definition of disturbances f(t,x,z), g(t,x,z),

and matrices A21, A22, K. With this control law, the system (3.27) is uniformly prac-

tically stable for (0, ] and the trajectories x and ultimately satisfy (Yue and Xu,

1996)

x O(), O().

Yue and Xu also proved the existence of the sliding motion for the sliding surface s =

BT1 P1x + BT2 P2 provided some condition are satisfied (Yue and Xu, 1996). Although,

their approach deals with disturbances and provides some certain robust characteristics,

it only guarantees the trajectories of the system travel in a O() boundary layer of the

origin. Furthermore, it is complicated to compute some parameters for the control law.

Su (1999) studied the problem of sliding surface design for the system (3.1). Like in

(Heck, 1991; Li et al., 1995a; Innocenti et al., 2003), the full-order system is separated

into slow and fast subsystems. Then, stabilizing state feedback controls are constructed

individually for each subsystem, leading to a composite control law

u = us + uf = K1x + K2z. (3.31)


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Then, the closed-loop system is written as

x

z

=

T11 T12

T21 T22

(3.32)

where

T11 = A11 + B1K1,

T12 = A12 + B1K2,

T21 = A21 + B2K1,

T22 = A22 + B2K2.

The closed-loop system is transformed into an exact slow and an exact fast subsystem

by using the Chang transformation (Chang, 1972; Kokotovic et al., 1986):

=

In1 HL H

L In2

x

z

= J

. (3.33)

The exact subsystems in the new coordinates are

= (T11 T12) = Ts

= (T22 + LT12) = Tf. (3.34)

There exist positive definite matrices Ps and Pf such that

PsTs + TTs Ps = Qs, Qs > 0

PfTf + TTf Pf = Qf, Qf > 0. (3.35)

Then, the sliding surface for the singularly perturbed system can be chosen as

s(x, z) = B1

B2/

T

JT Ps 0

0 Pf

J x

z

= 0. (3.36)

It was shown that (Su, 1999) if the sliding motion is achieved, the system is asymptoti-

cally stable. Like in (Yue and Xu, 1996), only one sliding surface is designed for the full

order system. However, a control strategy has not been provided to realize the sliding

motion.


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56

In this chapter, we address the problem of sliding mode control for a singularly

perturbed system with the external disturbance. While several papers in the literature

only address the stability of the closed-loop system (Heck, 1991; Li et al., 1995a; Inno-

centi et al., 2003), we consider both closed-loop stability and disturbance rejection. In

our method, a state feedback control law is firstly established to stabilize either slow or

fast dynamics. Then, a sliding mode control law is designed for the remaining dynamics

of the system to ensure stability and disturbance rejection. Putting the two controls

together produces a composite control law that makes the closed-loop system asymp-

totically stable. The advantage of the proposed method over others in the literature is

that external disturbances are completely excluded.

3.2 Problem Formulation

Consider a singularly perturbed system:

x(t)

z(t)

= A

x(t)

z(t)

+ Bu(t) + Df(t), (3.37)

A =

A11 A12

A21 A22

, B =

B1

B2

, D =

D1

D2

,

where x(t) Rn1 and z(t) Rn2 are the slow time and fast time state variables,u(t) Rm is the control input, is a small positive parameter. Matrix A22 is invertible,that is rank(A22) = n2. Matrices A, B, D are constant and of appropriate dimension.

Furthermore, f(t) Rr is an unknown but bounded exogenous disturbance with f(t) h.

Disturbance rejection and parameter variation invariance will be achieved if the

matching condition of Drazenovic (1969) is satisfied:

rank[B|D] = rankB. (3.38)

Due to this invariance condition, there exists an m r matrix G such that

D = BG. (3.39)


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57

Hence, D1 = B1G and D2 = B2G.

Our objective is to find a sliding mode control law to achieve both system stability

and disturbance rejection.

3.3 Main Results

In this section, we will present two sliding mode control strategies for the singularly

perturbed system (3.37). The two control methods share a similar procedure:

A state feedback control law is designed to maintain either the fast or slow modesasymptotically stable.

A discontinuous sliding mode control law for the remaining modes is establishedto reject disturbances.

The results are synthesized in a composite control law to ensure the stability androbustness of the whole system.

Before proceeding to the main results, we need the following assumption.

Assumption 3.1. (A0, B0) and (A22, B2) are controllable.

A0 = A11 A12A122 A21, B0 = B1 A122 B2.

This assumption allows us to construct state feedback control laws separately for

the slow and fast subsystems provided the original full-order system is controllable.

3.3.1 Dominating Slow Dynamics Approach

In this approach, a linear state feedback control law is designed to place eigenvalues

of the fast subsystem into appropriate positions, and then a sliding mode control law

is used for the slow subsystem to exhibit the desired slow time system performance.

Although the original singularly perturbed system can be decoupled into two time-

scales and into two lower dimensional state vectors, (t) and z(t), the similar type of

decomposition does not hold for the control law and the disturbance. As it can be


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58

seen in the slow and fast subsystems, the control law u(t) is decomposed only in the

time scale, us(t) and uf(t), but not in its dimension. In other words, both us(t) and

uf(t) are still m-vectors as its composite version u(t) = us(t) + uf(t). The same holds

for the disturbance f(t). Therefore, we allege that when it comes to the subject of

disturbance rejection, we can only have one subsystem that enters the sliding mode

unless an appropriate dimensional decomposition in the control law can be achieved.

With Assumption 3.1, a continuous fast-time state feedback control law is chosen

as

uf(t) = Kfz(t) (3.40)

such that A22 + B2Kf is asymptotically stable. The gain matrix Kf can be chosen

appropriately via the eigenvalue placement technique since (A22, B2) is controllable.Then, the system under the composite control law u(t) = us(t) + uf(t) is defined as

x(t)

z(t)

=

A11 (A12 + B1Kf)

A21 (A22 + B2Kf)

x(t)

z(t)

+

B1

B2

(us(t) + Gf(t)). (3.41)

By the change of variables

(t) = x(t) M()z(t), (3.42)

the system (3.41) is transformed into a lower triangular form (sensor form) (Kokotovic

et al., 1986)

(t)

z(t)

=

As 0

A21 A22 + B2Kf + A21M

(t)

z(t)

+

Bs

B2

(us(t) + Gf(t))

(3.43)

where

As = A11 M A21, (3.44)

Bs = B1 MB2 (3.45)and M is the solution to the following algebraic equation (Kokotovic et al., 1986)

A12 + B1Kf M(A22 + B2Kf) + A11M MA21M = 0. (3.46)

Matrix M can be found either using the fixed-point iterations or the Newton method

(Grodt and Gajic, 1988; Gajic and Lim, 2001).


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The system formulation (3.43) is related with its original form (3.37) via state

feedback (3.40) and the similarity transformation. Therefore, the controllability of the

slow subsystem pair (As, Bs) is intact (Chen, 1998). The design objective of the slow-

time control law us(t) is to stabilize the slow subsystem and simultaneously reject the

disturbance f(t) by applying the sliding mode control technique. We choose a sliding

surface for the dominating slow dynamics using the method of (Utkin and Young, 1979)

as

ss(t) = Cs(t) (3.47)

If m < n1, there exists a transformation Ts Rn1n1 for the slow subsystem of (3.43)such that (Utkin and Young, 1979)

TsBs =

0

Bs0

(3.48)

Under this transformation, the slow subsystem of (3.43) becomes

1(t)

2(t)

=

As11 As12

As21 As22

1(t)

2(t)

+

0

Bs0

(us(t) + Gf(t)) (3.49)

where

1(t)2(t)

= Ts(t).

The variable 2(t) should be regarded as a control input to the dynamic equation

of 1(t). According to (Utkin and Young, 1979), the controllability of (As, Bs) implies

the controllability of (As11, As12). As a result, we can find a gain matrix K1 such that

As11 As12K1 is stable. Then, a sliding surface can be chosen as

ss(t) = K11(t) + 2(t) = 0 (3.50)

Representing the sliding surface in (t) coordinates, we obtain

ss(t) = [K1 In1]Ts(t) = Cs(t) = 0 (3.51)

where Cs = [K1 In1]Ts.


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We are now in a position to design a sliding control law for the sliding surface (3.51)

as Cs is chosen. Taking the derivative of the sliding surface (3.51), we have

ss(t) = CsAs(t) + CsBsus(t) + CsBsGf(t). (3.52)

According to (Utkin and Young, 1979), CsBs is nonsingular ifBs has full rank. Assume

that Bs has full rank. Based on the control method of (Utkin, 1978), we choose a sliding

mode control law us(t) as

us(t) = (CsBs)1(CsAs(t) S1ss(t) + (1 + 1) ss(t)ss(t)) (3.53)

where 1 = CsBsGh, 1 is a positive parameter and matrix S1 is asymptoticallystable. The reaching condition is satisfied since

sTs (t) ss(t) = 1

2sTs (t)P1ss(t) 1ss(t) 1ss(t)

+ sTs (t)CsBsGf(t) < 1ss(t) (3.54)

where

P1 = ST1 S1 > 0. (3.55)

This means that us(t) is able to drive the slow variable (t) to reach the sliding surface

ss

(t) in a finite time and reject the disturbance f(t). In the following, we will estimate

the interval of the reaching time.

Choose a Lyapunov function

V(t) = sTs (t)ss(t). (3.56)

We have

max{P1}V(t) 2(1 + 21)

V(t) V(t) min{P1}V(t) 21

V(t). (3.57)

Let 1 be the time needed to reach the sliding mode (V(1) = 0). Taking the derivative

of (3.57), we have

2

max{P1} ln(max{P1}

V(0) + 21 + 41

21 + 41) 1

2

min{P1} ln(min{P1}

V(0) + 21

21). (3.58)


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In other words, the reaching time of the sliding mode lies in the interval

2

max{P1} ln(max{P1}

sTs (0)ss(0) + 21 + 4121 + 41

) 1

2min{

P1}

ln(min{P1}

sTs (0)ss(0) + 212

1

). (3.59)

Remark 3.1. The sliding mode control law (3.53) offers a flexibility in adjusting the

reaching time. Inequalities (3.59) show that choosing appropriate candidates for S1

(P1), 1, and 1 affects the reaching time. Since 1 and 1 constitute the magnitude of

the control effort in sliding mode, their large values are undesired. Hence, we only need

to pick up a suitable value of S1 (P1) to obtain a fast reaching time.

From (3.40) and (3.53), the composite control is given by

u(t) = Kfz(t) (CsBs)1(CsAs(t) S1ss(t) (1 + 1)ss(t)

s(t)). (3.60)

In terms of the original state variables, the control law is rewritten as

u(t) =Kfz(t) (CsBs)1(CsAs(x(t) M z(t)) S1Cs(x(t)

Mz(t)) (1 + 1) Cs(x(t) M z(t))Cs(x(t) M z(t))). (3.61)

When the sliding mode is achieved, one can use the equivalent control method

(Utkin, 1977) to study the dynamics of the closed-loop system. The stability of the

closed-loop system is guaranteed by the following theorem.

Theorem 3.1. There exists > 0 such that, in the sliding mode, the closed-loop

system is asymptotically stable for (0, ] and invariant to the external disturbancef(t).

Proof. To study the dynamics of the closed-loop system under the control law (3.60)

or (3.61), we employ the equivalent control method of (Utkin, 1977). The equivalent

control of the sliding motion is defined by solving ss(t) = 0. This yields

ueqs (t) = (CsBs)1(CsAs(t) + CsBsGf(t)). (3.62)

Substituting the equivalent control into (3.43) results in the following equivalent dy-

namics (t)

z(t)

= 1

(t)

z(t)

(3.63)


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where

1 =

As Bs(CsBs)1CsAs 0

A21 B2(CsBs)1CsAs A22 + B2Kf + A21M

.

The dynamics of the system (3.63) is defined by the eigenvalues of AsBs(CsBs)1CsAsand A22+B2Kf+A21M. Matrix AsBs(CsBs)1CsAs contains m zeros correspondingto the sliding motion and n1 m stable eigenvalues. Since matrix A22 + B2Kf isasymptotically stable, there exists a small 0 such that for all [0, ] theeigenvalues of A22 + B2Kf + A21M have negative real parts. As a result, the closed-

loop system is asymptotically stable according to (Utkin, 1977).

Remark 3.2. One can only study the stability of the slow dynamics by applying the

equivalent control to the dynamics equation (3.49). However, according to (3.43), the

whole transformed system is still affected by us. Hence, when proving stability of the

closed-loop system, we still need to consider the influence of us on the whole system.

3.3.2 Dominating Fast Dynamics Approach

This approach presents a composite control law that consists of slow state feedback

control and fast sliding mode control. According to Assumption 3.1, (A0, B0) is con-

trollable. Thus, there exists a gain matrix Ks such that state feedback control

us(t) = Ksx(t) (3.64)

renders matrix A0 + B0Ks asymptotically stable. The system under the control law

u(t) = us(t) + uf(t) is described as

x(t)z(t)

=A11 + B1Ks A12

A21 + B2Ks A22

x(t)z(t)

+B1

B2

(uf(t) + Gf(t)). (3.65)

Introducing the change of variables

(t) = z(t) + L()x(t) (3.66)


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the system (3.65) is brought into the actuator form (Kokotovic et al., 1986)

x(t)

(t)

=

A11 + B1Ks A12L A12

0 Af

x(t)

(t)

+

B1

Bf

(uf(t) + Gf(t))

(3.67)

where

Af = A22 + LA12 (3.68)

Bf = B2 + LB1 (3.69)

and L is the solution to the following algebraic equation (Kokotovic et al., 1986)

A21 + B2Ks A22L + L(A11 + B1Ks) LA12L = 0. (3.70)

This equation can be solved using either the fixed-point iterations or the Newton method

(Grodt and Gajic, 1988). Since L = A122 (A21 + B2Ks) + O() (Kokotovic et al., 1986),

we have

A11 + B1Ks A12L = A0 + B0Ks + O() (3.71)

Because A0 + B0K is asymptotically stable, there exists a small > 0 such that for

all [0, ], A11 + B1Ks A12L is asymptotically stable.

We are now in a position to construct a sliding mode control law for the fast sub-system in (3.67). Employing the same technique as in the dominating slow dynamics

approach, we choose a sliding surface via the method of (Utkin and Young, 1979) as:

sf(t) = Cf(t) (3.72)

If m < n2, there exists a transformation Tf Rn2n2 for the fast subsystem of (3.67)such that Utkin and Young (1979)

TfBf = 0

Bf0

. (3.73)

Under this transformation, the fast subsystem of (3.67) becomes

1(t)

2(t)

=

Af11 Af12

Af21 Af22

1(t)

2(t)

+

0

Bf0

(uf(t) + Gf(t)). (3.74)


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Consider 2(t) as a control input to the dynamic equation of 1(t). Since (Af21, Af22)

is controllable, we can find a gain matrix K2 such that As21 As22K2 is asymptoticallystable. Then, a sliding surface can be chosen as

sf(t) = K21(t) + 2(t) = 0. (3.75)

Representing this sliding surface in the previous coordinates, we obtain

sf(t) = [K2 In2]Tf(t) = Cf(t) = 0 (3.76)

where Cf = [K2 In2]Tf.

Taking the derivative of the sliding surface (3.76) with respect to t, we have

sf(t) = CfAf(t) + CfBfuf(t) + CfBfGf(t). (3.77)

With the assumption that the disturbance f(t) is bounded, and Bf has full rank, a

control law uf(t) can be chosen as

uf(t) = (CfBf)1(CfAf(t) S2sf(t) + (2 + 2) sf(t)sf(t)) (3.78)

where 2 = CfBfGh, 2 is a positive parameter, and matrix S2 is asymptoticallystable. The reaching condition is satisfied since

sTf(t) sf(t) = 12 sTf(t)P2sf(t) 2sf(t) 2sf(t) + sTfCfBfGf(t) < 2sf(t)

(3.79)

where

P2 = ST2 S2 > 0. (3.80)

Similarly to (3.59), the reaching time of the sliding mode satisfies

2

max{P2}ln(

max{P2}

sT2 (0)s2(0) + 22 + 42

22 + 42)

2

2min{P2} ln(

min{P2}

sT2 (0)s2(0) + 22

22). (3.81)

Like in the first approach, the reaching time of the sliding mode can be monitored

by choosing a suitable value of P2 (S2) without affecting the magnitude of the control

effort during sliding.


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The composite control law is described in terms of the slow state feedback law and

the fast sliding mode control law as follows

u(t) =Ksx(t) (CfBf)1(CfAf(t) S2sf(t) (2 + 2) sf(t)

sf(t)

). (3.82)

In the original coordinates, the control law (3.82) is

u(t) =Ksx(t) (CfBf)1(CfAf(z(t) + Lx(t)) S2Cf(z(t) + Lx(t))

(2 + 2)Cf(z(t) + Lx(t))

Cf(z(t) + Lx(t))). (3.83)

The stability of the closed-loop system under the control law (3.83) is proved in the

following theorem.

Theorem 3.2. Assume(A, B) is controllable. Then there exists > 0 such that in the

sliding mode, the closed-loop system is asymptotically stable for (0, ] and invariantto the external disturbance f(t).

Proof. Like the dominating slow dynamics approach, we employ the equivalent control

method to study the dynamics of the closed-loop system. The equivalent control of the

sliding motion of (3.75) is defined by solving sf(t) = 0 as follows

ueqf (t) = (CfBf)1CfAf(t) Gf(t). (3.84)

Hence, the equivalent dynamics of the closed-loop system under the equivalent con-

trol law (3.84) is given by x(t)

(t)

= 2

x(t)

(t)

(3.85)

where

2 = A11 + B1Ks

A12L A12

B1(CfBf)1CfAf

0 Af Bf(CfBf)1CfAf

.

The dynamics of the system (3.85) is specified by the eigenvalues of A11 + B1KsA12Land 1 (Af Bf(CfBf)1CfAf). The eigenvalues of matrix Af Bf(CfBf)1CfAfinclude m zeros corresponding to the sliding motion and n1 m asymptotically stableeigenvalues. In addition, A11 + B1Ks A12L is asymptotically stable. As a result, the


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dynamics of the system (3.85) is represented by n1 + n2 m stable eigenvalues and mzeros. According to (Utkin, 1977), the closed-loop system is asymptotically stable. This

implies that

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Sliding Mode Control For Systems With Slow and Fast Modes

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